A rainstorm increased the amount of water stored in State J reservoirs from 124 billion gallons to 138 billion gallons....

1. Translate the problem requirements: We need to find how many gallons the reservoirs were short of total capacity before the storm. This means: (Total Capacity) - (Amount before storm) = Shortage before storm
2. Find total capacity using current state: Use the fact that 138 billion gallons represents \(\mathrm{82\%}\) of total capacity to calculate the full capacity
3. Calculate the shortage before the storm: Subtract the pre-storm amount (124 billion gallons) from the total capacity to find how much they were short

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're actually looking for in plain English. The problem asks: "how many billion gallons of water were the reservoirs short of total capacity prior to the storm?"

Think of it this way: imagine the reservoirs as a giant bucket. Before the storm, this bucket had 124 billion gallons. After the storm, it had 138 billion gallons. We know that after the storm, the bucket was \(\mathrm{82\%}\) full of its total capacity.

What we want to find is: How much more water could the bucket have held before the storm? In other words, what's the difference between the bucket's total capacity and the 124 billion gallons it had before the storm?

So our target calculation is: (Total Capacity of the reservoirs) - (Amount before storm) = Shortage before storm

Process Skill: TRANSLATE - Converting the problem language into a clear mathematical relationship

2. Find total capacity using current state

Now we need to figure out the total capacity of the reservoirs. We know that after the storm, the reservoirs contain 138 billion gallons, and this represents \(\mathrm{82\%}\) of the total capacity.

Think about it this way: if 138 billion gallons is \(\mathrm{82\%}\) of the bucket, then what is \(\mathrm{100\%}\) of the bucket?

We can set this up as a simple proportion: if 138 billion gallons represents \(\mathrm{82\%}\), then the total capacity represents \(\mathrm{100\%}\).

Using basic percentage relationships:
\(\mathrm{82\% \text{ of Total Capacity} = 138 \text{ billion gallons}}\)

To find the total capacity:
Total Capacity = \(\mathrm{138 ÷ 0.82}\)
Total Capacity ≈ 168.3 billion gallons

Let's verify this makes sense: \(\mathrm{168.3 × 0.82 ≈ 138}\) ✓

3. Calculate the shortage before the storm

Now we can find how much the reservoirs were short of total capacity before the storm.

We know:

• Total capacity ≈ 168.3 billion gallons
• Amount before storm = 124 billion gallons

The shortage before the storm = Total Capacity - Amount before storm
Shortage = \(\mathrm{168.3 - 124 = 44.3}\) billion gallons

Since we're looking for an approximate answer, this rounds to 44 billion gallons.

4. Final Answer

The reservoirs were short of total capacity by approximately 44 billion gallons prior to the storm.

Looking at our answer choices:

1. 9
2. 14
3. 25
4. 30
5. 44

Our calculated answer of 44 billion gallons matches choice (E) exactly.

Answer: (E) 44

Common Faltering Points

Errors while devising the approach

1. Misinterpreting what "shortage" means

Students often confuse what the question is asking for. They might think the shortage is simply the 14 billion gallon increase from the storm (\(\mathrm{138 - 124 = 14}\)), rather than understanding that shortage means the difference between total capacity and the amount before the storm. This leads them to select answer choice (B) 14.

2. Incorrectly setting up the relationship between current state and total capacity

Some students struggle with the percentage relationship and might incorrectly think that 124 billion gallons (the amount before the storm) represents \(\mathrm{82\%}\) of total capacity, rather than understanding that 138 billion gallons (after the storm) represents \(\mathrm{82\%}\) of total capacity. This fundamental misunderstanding leads to calculating the wrong total capacity.

Errors while executing the approach

1. Making arithmetic errors in the division step

When calculating \(\mathrm{138 ÷ 0.82}\), students might make computational errors, especially if they try to do this division mentally or convert \(\mathrm{82\%}\) incorrectly. Some might use 82 instead of 0.82, leading to a drastically wrong total capacity of \(\mathrm{138 ÷ 82 ≈ 1.68}\) billion gallons, which doesn't make logical sense.

2. Rounding errors or premature rounding

Students might round the total capacity calculation too early (for example, rounding 168.3 to 168) and then carry this error through to the final calculation, potentially leading them to select a different answer choice like (D) 30 instead of the correct answer.

Errors while selecting the answer

No likely faltering points - The final calculation step is straightforward subtraction, and the result clearly matches one of the given answer choices without requiring additional interpretation or conversion.